Compatible Feigin-Odesskii Poisson brackets

TL;DR

This paper characterizes when Feigin-Odesskii Poisson brackets related to elliptic curves are compatible, linking compatibility to geometric configurations like scrolls or Veronese surfaces, and analyzes specific non-compatible cases.

Contribution

It provides a complete characterization of compatibility conditions for Feigin-Odesskii Poisson brackets in terms of geometric embeddings and determines specific quartic relations in a special case.

Findings

Compatibility occurs iff brackets are in a scroll or Veronese surface.

Identifies an exception in the case n=3.

Determines the quartic for non-compatible brackets in the n=3 case.

Abstract

We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$). In the case $n=3$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $E_1$ and $E_2$.